Partiality, Revisited
The Partiality Monad as a
Quotient Inductive-Inductive Type
Thorsten Altenkirch1 Nils Anders Danielsson2
Nicolai Kraus1
1
University of Nottingham 2 University of Gothenburg
FoSSaCS, 26 April 2017
Partiality
Task: Given f ∶ N → N, find n ∶ N such that f (n) = 0.
In many languages (e.g. Haskell):
easy to write such a function (N → N) → N
Our setting: intensional Martin-Löf type theory (e.g. Agda)
● formal system with Σ-, Π-, identity types, . . .
● can be used for programming
● potential foundation of mathematics
All functions are total!
Partiality
Our goal: A monad
M ∶ Type → Type
such that M(A) is a “type of partial elements” in MLTT.
Attempt: Maybe monad, M(A) ∶≡ 1 + A.
Not good, “too decidable”:
cannot construct suitable function (N → N) → 1 + N.
Attempt: M(A) ∶≡ ΣQ∶Prop (Q → A).
(Cf. Escardó-Knapp 2017)
Here not good, “too undecidable”.
Our goal: something “semidecidable”.
Delay monad
Better attempt: Delay Monad (Capretta 2005).
D(A) is the coinductive type generated by
● now ∶ A → D(A)
● later ∶ D(A) → D(A).
Equivalent representation: functions N → (1 + A) which
become constant once they are inr(a).
Back to the problem “find a zero”:
Yes, we can define a function (N → N) → D(N).
But D(A) is very intensional:
later(now(a)) =/ later(later(now(a)).
Our goal: more extensionality.
Delay monad, quotiented
Weak bisimilarity: binary relation ≈ on D(A).
Intuition: x ≈ y iff x and y become equal after removing
some “laters”, thus: later(now(a)) ≈ later(later(now(a))).
Chapman, Uustalu, Veltri:
Quotienting the delay monad by weak bisimilarity (2015).
Use D(A)/ ≈ (quotient as introduced by Hofmann).
D(_)/≈ is a monad on Type assuming countable choice.
Countable choice: Πn∶N ∥A(n)∥ → ∥Πn∶N A(n)∥,
“for every n, there exists A(n)” → “there exists a function
giving A(n) for every n”.
Our goal: avoiding choice.
Quotient inductive-inductive types
We use a combination of two concepts:
● higher inductive types from homotopy type theory:
inductive types can have constructors for equalities
● induction-induction
This combination is also used in the HoTT book to define
the Cauchy reals.
● Caveat: computational interpretation conjectured, but
still experimental
● Only need special case (quotient inductive-inductive
types), examined by Dijkstra (2016).
Partiality monad, the construction
Define type (set) A– and ⊑∶ A– → A– → Prop simultaneously:
Inductive type (set) A– with constructors:
η ∶ A → A–
– ∶ A–
⊔ ∶ (Σs∶N→A– Πn∶N sn ⊑ sn+1 ) → A–
α ∶ Πx,y ∶A– x ⊑ y → y ⊑ x → x = y
Inductive relation ⊑ given by the rules:
x ⊑y
x ⊑x
y ⊑z
x ⊑z
–⊑x
Πn∶N sn ⊑ x
Πn∶N sn ⊑ ⊔(s, p)
⊔(s, p) ⊑ x
Further characterisations
We get an adjunction:
our construction
SET
–
ωCPO
forgetting structure
Note: categories can be defined internally:
● SET – types (actually sets)
● ωCPO – types (sets) with structure making them
ω-complete partial orders
Connection between the constructions
Assuming countable choice, our construction is equivalent to
Chapman-Uustalu-Veltri’s:
A– ≃ D(A)/≈
Why do we need countable choice?
! w ∶ D(A) → A
! w preserves weak bisimilarity (≈)
! (w (d) = w (e)) → (d ≈ e)
–
! surjectivity: Πx∶A– ∥Σd∶D(A) w (d) = x∥
Induction on x; case x ≡ ⊔s needs countable choice
Final words
Some applications:
● non-terminating functions as fixed points,
e.g. (N → N) → N–
● functions from the Cauchy reals, developed further by
Gilbert (2017)
● topology with 1– , . . .
We have formalised this in Agda.
Take home message:
Constructing an inductive type simultaneously with its
equalities is also useful “outside homotopy type theory”.
Thank you for your attention!